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\newcommand{\np}{\mathop{\rm NP}}
\newcommand{\conp}{\mathop{\rm co-NP}}
\newcommand{\atisp}{\mathop{\rm ATISP}}
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\begin{document}
\noindent {\large Advanced Complexity Theory \hfill
Madhu Sudan\\[-.1in]
6.841/18.405J \hfill \mbox{}\\[-.1in]
Due: Monday, March 18, 2002\hfill \mbox{}\\[.4in]}
{\LARGE \centering Problem Set 1 \\[.4in] \par}
\section*{Problems}
\begin{enumerate}
\item Give a randomized logspace algorithm for determining if
an undirected graph is bipartite (i.e., if the vertices can be partitioned
into two sets $V_1$ and $V_2$ such that all edges have exactly one
endpoint in $V_1$ and one in $V_2$). Include a formal definition of
the language you are working with, and say whether your algorithm
places this language in BPL, RL or co-RL.
For extra credit, show that this problem is in ZPL.
\item A language $L$ is said to be {\em rare} if there exists
a polynomial
$p$ such that for every $n$, the number of strings in $L$ of length
$n$ is at most $p(n)$.
Show that if a rare language is NP-complete then the hierarchy
collapses.
For extra credit, show that if a rare language is NP-complete then
NP=P.
\item The depth of a ciruit is the longest path from an input node
to an output node in the circuit. A circuit is said to be a formula
is every computation node has fan-out one (i.e., only one wire
leaves every internal node). In this problem, we'll only
consider formulae
and circuits with unary NOT gates, and binary AND and binary OR gates.
Show that a family of functions $\{f_n:\{0,1\}^n \to \{0,1\}\}_n$
has a family logarithmic depth circuits if and only if it has
the family of functions has polynomial size formulae.
\item
If PSPACE $\subseteq P/_{\poly}$ then show that PSPACE $=\Sigma_k^P$
for some constant $k$.
\newcommand{\piyes}{{\Pi_{\rm YES}}}
\newcommand{\pino}{{\Pi_{\rm NO}}}
\item A promise problem is a generalization of the notion of a
language. A promise problem $\Pi = (\piyes,\pino)$
consist of two disjoint subset
$\piyes$ and $\pino$ of $\{0,1\}^*$. A Turing machine $M$ is
said to decide $\Pi$ if it accepts every string in $\piyes$ and
rejects every string in $\pino$ (and we don't care what happens
with the rest). Promise problems play an important role in
modern complexity theory.
Let promise-RP be the class of promise problems decided by a
randomized polynomial Turing machine $M(\cdot,\cdot)$: i.e.,
if $x \in \piyes$ then $\Pr_y[M(x,y)$ accepts$] \geq 2/3$,
and if $x \in \pino$ then $\Pr_{y}[M(x,y)$ accepts$] = 0$.
Show that if P=promise-RP, then P=BPP.
I.e., suppose, for every promise problem $\Pi$ in promise-RP,
there is polynomial time algorithm $A$ that accepts every
instance in $\Pi_{\rm YES}$ and rejects every instance
in $\Pi_{\rm NO}$, then show that P=BPP.
\end{enumerate}
\noindent {\bf Instructions:}
\begin{itemize}
\item Usual rules on collaboration and references.
\item Turn in the solutions to the above problems by 11am on
Monday, March 18, 2002 (even though we have no lecture then).
\end{itemize}
\newpage
\section*{Additional Exercises: Not to be turned in!!}
The following exercises are highly recommended. They
give more practice with the notions in class.
Doing the exercises is not mandatory.
\begin{enumerate}
\item Show that ZPP = RP $\cap$ co-RP.
\item For a deterministic polytime Turing machine $M$
and polynomial $p$,
define a sequence of strings $(a_1,\ldots,a_n)$ to be
GOOD if $|a_i| = p(i)$ and for every cnf formula
$\phi$ of length $i \leq n$, $\phi \in SAT \Leftrightarrow
M(\phi,a_i)$ accepts.
Use the self-reducibility of SAT to show that the
task of recognizing if a sequence of strings is GOOD
is in co-NP.
\item A Branching Program is yet another non-uniform model
of computation. Such a program is specified by a directed
acyclic graph on $m$ nodes with one designated start node
and two end nodes labelled $0/1$. Start nodes and all other
internal nodes are labelled by a variable name from
$x_1,\ldots,x_n$. End nodes have out-degree $0$. All other
nodes have out-degree two with one edge labelled $0$ and
the other labelled $1$. The branching program represents
a Boolean function as follows: Given a 0/1 assignment to
the variables we follows the path out of the start node
that is consistent with the assignment (i.e,, taking the
edge labelled 0 out of a vertex labelled $x_i$ if $x_i = 0$
etc.) till we reach an end node. The label of the end node
gives the value of the function on this input. The size of
a branching program is the number of nodes in it. The size
represents ``non-uniform'' space requirement of a computation.
\begin{enumerate}
\item If $L \in $SPACE$(s(n))$ then show that $L_n$ is
computed by a poly$(2^{s(n)})$-sized branching program.
\item Given a branching program, show that it is NP-complete
to determine if the branching program accepts any input.
\item A branching program is read-once if every variable
appears at most once on every input/output path. Give a
randomized algorithm to test if two read-once branching programs
accept the same Boolean function.
\end{enumerate}
\end{enumerate}
\end{document}